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What is a vector?

When students first meet vectors they are pretty much told that vectors are arrows. They move arrange the arrows head-to-tail to add them, and they lengthen the arrows when they multiply them by numbers. Sometimes the vector is represented using coordinates, but they are told that this is shorthand for the arrow that goes from the origin to that point. Lovely.

And then, in first year maths at uni, they are told that vectors are a list of coordinates. To add them, you add the coordinates, and to multiply them by a number, you multiply each coordinate. You create sets of vectors using equations, which are represented as coloured regions in space – a point is coloured in if its coordinates satisfy the equation. The only way to make sense of this is to think of your set of coordinates as a point, not an arrow. Oh dear.

So which is it then? Are vectors points or arrows?

There are at least four answers to this:

1. Vectors are points, and the arrow picture is simply a way to helps us visualise what the result will be when we add them and multiply them by numbers. So when I see a set of coordinates, I think of the point in space they represent, and when I draw a set of them, I just colour in the points. And when I add them I am just using the rules for how they add to find the answer, but I can visually picture where in space the answer will be by imagining them as arrows.

2. Some vectors are points, and other vectors are arrows, depending on context. So when you add two vectors, the first one is a location – a place to start, if you will – and the other one is an arrow – a direction to go. The result is a the place you get to by starting at the point and moving the arrow, and is of course a point again.

3. Vectors live in two separate worlds, one where they are points, and another where they are arrows. There is a world where vectors represent locations in space and they happily find themselves inside our outside sets. But there is another world, where vectors are arrows and they happily arrange themselves head-to-tail when they want to add. When you have two points, and you want to add them, they quickly move over into arrow-world, do their adding, and then the answer comes back as a point again.

4. Vectors are neither points nor arrows. They are mathematical objects with no innate physical reality at all. The concept of drawing a vector as a point or an arrow is simply a way to represent them on paper and it is not what they actually ARE.

And which answer do I think is the right one? All of them, none of them – this sort of discussion is philosphy, not maths, and in philosophy there's no right or wrong. You pick what works for you. Indeed, being able to see it from different viewpoints only enriches your understanding.

Still, I do quite like the "separate worlds" idea.

This comment was left on the original blog post: 

David Roberts 23 July 2013:

I think the main difference is that a vector as an arrow is an inherently geometric idea: you draw them, move them around the page and so on. The list of numbers is more algebraic, and more static: you can’t pick up (2,5,1) and stick it on the end of (22,7,0) like you can with those vectors represented as arrows.

There are really different things going on when you move vectors as arrows around the page that don’t happen in the list of numbers approach. The trick is that those concepts don’t get formalised for several years after R^n is formally introduced, and are usually couched in a much more complicated and abstract setting.